Matrix norms and search for sterile neutrinos
aa r X i v : . [ h e p - ph ] A p r Matrix norms and search for sterile neutrinos
Wojciech Flieger ∗ , Franciszek Pindel, Kamil Porwit Institute of Physics, University of Silesia, Katowice, PolandE-mails: [email protected] , [email protected] , [email protected] Matrix norms can be used to measure the "distance" between two matrices which translates natu-rally to the problem of calculating the unitary deviation of the neutrino mixing matrices. Varietyof matrix norms opens a possibility to measure such deviations on different structural levels ofthe mixing matrix.
Corfu Summer Institute 2018 "School and Workshops on Elementary Particle Physics and Gravity"(CORFU2018)31 August - 28 September, 2018Corfu, Greece ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). https://pos.sissa.it/ atrix norms, neutrino mixings
Wojciech Flieger
1. Introduction
Neutrino experiments do not exclude that additional right-handed sterile neutrinos exist. Thereare many ways, inspired by theory, in which they can be discovered both in neutrino oscillations[1, 2, 3, 4, 5, 6, 7] and collider physics [8, 9, 10, 11, 12, 13]. Their masses are practically notlimited, ranging from eV to TeV, or the Planck scale. Experiments and theoretical studies aredevoted to search for such signals. Interestingly, some present experiments and the experimentalsignals suggest that the fourth type of neutrino may exist.Namely, measurements of the ¯ ν e flux at small distances from nuclear reactors gives 6% lessevents than expected [14]. Such a deficit referenced as a "Reactor Antineutrino Anomaly" canbe explained as active-sterile antineutrino oscillations at very short baselines [15]. Moreover, theNEUTRINO-4 reactor experiment claims to detect an electron antineutrino to sterile neutrino os-cillation at the 3 σ significance level [16]. Most probable values for oscillation parameters areestimated as sin θ = .
39 and ∆ m = .
34 eV .Gallium solar neutrinos experiments also observe less events than predicted. Deficit in data isreported by GALEX and SAGE collaborations [17]. Statistical significance for such "GalliumAnomalies" in terms of neutrino oscillations was recently estimated at the level of 3 σ [18].Clues for sterile neutrino oscillations exist also in the long baseline experiments. First reported ab-normalities were reported by the LSND collaboration in 1995. More efficient analysis from 1996showed an excess of electron antineutrinos events from the muon antineutrino beam [19]. However,some publications cast in doubt analysis done within the LSND and its interpretation in favour ofsterile neutrinos. Reduction of the significance level below 3 σ is claimed [20, 21]. Recent resultsfrom the MiniBooNE experiment show a massive excess of electron (anti)neutrino events from themuon (anti)neutrino beam. Surplus of (79)281 events in correspondence to predicted number ofevents was reported [22].However, discrepancies between predicted and observed data events can be explained not onlyby adding new neutrinos to the existing theory. There are still many exciting possibilities suchas: resonant neutrino oscillations [23, 24, 25], Lorentz violation [26, 27], sterile neutrino decay[28, 29] and many more.Here we will focus on the theoretical analysis connected with deviations from unitarity ofthe standard mixing matrix U PMNS . Such deviations would be a clear signal of known neutrinosmixing with additional right-handed sterile states. We extend this kind of analysis and study howdeviations from unitarity can be defined in the framework of advanced matrix algebra using matrixnorms. The advantage of using norms instead of the maximal absolute value of the matrix is ofmathematical nature - matrix norms in a natural way reflects properties of matrices. Moreover, theyposses important properties which help with theoretical calculations and manipulation of matrices.Most of the background for such analysis was established in [30] where a notion of singular values,matrix norms and contraction was introduced. This machinery allows us to determine when a non-unitary mixing matrix can describe physical phenomenon. Moreover, it is possible to define a set ofall physically admissible mixing matrices. On top of that the method of so-called unitary dilationgives us the possibility to study in a systematic way scenarios with additional neutrinos [31].1 atrix norms, neutrino mixings
Wojciech Flieger
2. Parametrizations of the neutrino mixing matrix
To describe neutrino non-standard mixings usually two parameterizations are used, commonlydenoted by α and η [32, 33, 34, 35], defined in the following way U PMNS = ( I − η ) V , U PMNS = ( I − α ) W . (2.1)Here η is a Hermitian matrix, α is a lower triangular matrix and U , V and W are unitary matrices.The size of the deviation is described by matrices α and η . In the limit η , α →
0, the standard U PMNS unitary matrix is restored [36, 37, 38]. In contrast to these approaches we want to findan optimal unitarity deviation one-parameter indicator. This value will be estimated using matrixnorms which have the property of functions to measure the "distance" between two matrices. Wewill show an example of differences between norms when applied to the neutrino data.
3. Matrix norms and deviations from unitarity
Let us recall the general definition of the matrix norm. We will use it to study a deviation fromunitarity of the mixing matrix.
Definition 1.
A matrix norm is a function k · k from the set of all complex matrices into R whichfor any A , B ∈ C n × n and any α ∈ C satisfies the following conditions:1. k A k ≥ and k A k = ⇔ A = k α A k = | α |k A k k A + B k ≤ k A k + k B k k AB k ≤ k A kk B k As in the case of the "normal" vector norms there are plenty of matrix norms, each useful fora different application. We will not discuss here all of them but it is important to mention that allof them are equivalent which means that for a given matrix A ∈ C n × m any two matrix norms k · k a and k · k b satisfy k k A k a ≤ k A k b ≤ s k A k a (3.1)where k , l are positive numbers.Specific matrix norms are used in many different applications, especially in data analysis,economics, optimization and image processing [39, 40, 41, 42].2 atrix norms, neutrino mixings Wojciech Flieger
Some representative examples of the matrix norms are the following1. Operator norm (ON): k A k = max k x k = k Ax k = σ max ( A ) . (3.2)2. Frobenius norm (FN): k A k F = q Trace ( A † A ) = s n ∑ i = m ∑ j = | a i j | = vuut min { n , m } ∑ i = σ i ( A ) . (3.3)3. Maximum absolute column sum norm (MACN): k A k = max j m ∑ i = | a i j | . (3.4)4. Maximum absolute row sum norm (MARN): k A k ∞ = max i m ∑ j = | a i j | . (3.5)Now we are ready to define the way to measure the deviation from unitarity of the mixing matrix. Definition 2.
Let A ∈ C n × n be a given matrix and let k · k be any matrix norm then the function k I − AA † k (3.6) measures how far A is from the unitary matrix.
4. Structure of the physical region for neutrino mixing matrices
Not all matrices that deviate from unitarity can describe physical mixing. Whether a matrixcan be treated as physical is determined by the probability of transition between flavour states. Inoscillations we cannot lose neutrinos which means that the probability of transition from a givenstate to any other must be one. This is ensured by the unitarity of the mixing matrix. Thus phys-ically admissible mixing matrices must be either unitary or can be extended to the unitary matrix.Such matrices are known as contractions, i.e., the matrices with the largest singular value less orequal to one, symbolically σ max ( A ) ≤
1. On the other hand we have bounds for mixing parametersobtained from experiments. These two requirements together give us the definition of the physicalregion. A region of physically admissible mixing matrices is defined as a convex hull spanned onall 3-dimensional unitary matrices with parameters bounded by experiments or equivalently as aset of all convex combinations of these unitary matrices. Thus the formal definition looks in thefollowing way Ω = { M ∑ i = α i U i | U i ∈ U × , α , ..., α M ≥ , M ∑ i = α i = , θ , θ , θ and δ restricted by experiments } . (4.1)It can also be seen as an intersection of the unit ball with respect to the Operator norm with thehyperrectangle determined by experimental ranges. The formal definition of the physical region Ω opens a possibility for further interesting studies and better understanding of the neutrino mixing.The first issue is the shape of the physical region. Since the mixing matrices are 9-dimensionalobjects (real or complex) the exact shape of Ω is not possible to visualize. However we could gain agreat chunk of information by projecting this region onto two or three dimensional subspaces. Thisproblem can be solved by incorporating methods of Semidefinite Programming (SDP) [43, 44]. The3 atrix norms, neutrino mixings Wojciech Flieger next issue concerns possible extensions of the Standard Model by introducing sterile neutrinos. Itis known that the number of singular values strictly less than one controls the smallest possibleunitary extension of the 3-dimensional mixing matrix, i.e. the number of additional neutrinos[30]. Thus the collection of matrices that allow the smallest unitary dilation with one, two or threesterile neutrinos divides the region Ω into three disjoint subsets. It is interesting how these subsetsare distributed within Ω . Are they distributed randomly or they clustered? Visualization of thephysical region mentioned before can also be helpful. The next important issue is the entrywisecharacterization of contractions. This will allow to construct matrices from Ω in a simple way andgive entrywise description of this region. There are more open issues about the region of physicallymixing matrices like: what is the minimal number of unitary matrices necessary to cover the wholeregion.
5. A numerical example
In what follows, let us construct a matrix from the region Ω as a convex combination of unitary U PMNS matrices and determine its deviation from unitarity using different norms introduced before.All calculations have been done with 16 digits precision. However, we will keep 4 significant digitsfor all numbers given, as an accuracy which can be achieved for error estimation (0.003) in suchanalysis is at the same level [30]. The matrix V from the physical region Ω is constructed as theconvex combination of 3 unitary U PMNS matrices characterized by the following sets of mixingparameters: U : θ = . , θ = . , θ = . , U : θ = . , θ = . , θ = . , U : θ = . , θ = . , θ = . . (5.1)Thus the matrix V is the following sum V = α U + α U + α U , (5.2)where the coefficients are α = . , α = . , α = . . (5.3)Such a convex combination gives the following mixing matrix V = . . . − . . . . − . . . (5.4)Let us first check if this matrix is a contraction. To do this we calculate a set of its singular values σ ( V ) = { . , . , . } . (5.5)The largest singular value is less than one, therefore the matrix V is a contraction. Although theresult can suggest that taking an error into account the σ ( V ) can possible take the value above4 atrix norms, neutrino mixings Wojciech Flieger one, this is not the case since the construction (4.1,5.2) impose the contraction property. Now letus study deviation from unitarity for this matrix using functions introduced in the Definition 2. k I − VV † k = . , k I − VV † k F = . , k I − VV † k = . , k I − VV † k ∞ = . , max | I − VV † | = . . (5.6)We can see that for the considered example, besides MACN: k · k and MARN: k · k ∞ definedin (3.4, 3.5), other norms give different results. MACN and MARN norms give the same resultbecause we calculate unitarity deviations for the symmetric matrix VV † . The definitions (3.4, 3.5)suggest that these two specific norms can be used to measure in a simple way the largest unitarydeviation among mixings with particular flavor or massive states. Unfortunately, this is not the casebecause in the product VV † these states are mixed. However, information about unitarity deviationson the level of flavors states is gathered by the diagonal elements of VV † . Similarly, informationabout unitarity deviations on the level of massive states is given by the diagonal elements of V † V . Itis necessary to check whether these elements satisfy requirements of the matrix norm. If not, theyare still attractive functions to use in the case of neutrinos as the measure of the unitarity deviationon the levels of flavour and massive states but additional numerical and mathematical studies arenecessary. If we are interested in the absolute deviation from unitarity the Frobenius norm shouldbe the proper choice as it takes the whole structure of the matrix into account.
6. Summary
Matrix theory and matrix analysis are fruitful areas of mathematics with a broad range ofapplications. They can also be used to give new insights into neutrino mixings analysis [30]. Herewe have discussed a notion of matrix norms and their application in the calculation of deviationfrom unitarity of the neutrino mixing matrix. In general, different norms give different numericalestimates. We have made a rough estimate which one would be optimal for the unitary deviation ofthe neutrino mixing matrix. The example shows that the Frobenius norm gives the largest numericalvalue of the deviation. This agrees with the definition (3.3) because it takes into account all entriesof the matrix. That is why the Frobenius norm seems to be a right choice to be taken if the overallamount of the unitarity deviation of the mixing matrix is discussed. The usual way of calculatingthe unitary deviation using the maximum of absolute value is very close to the result given by theOperator norm (3.2). In the current form of the Definition 2 (3.6) the MACN and MARN norms(3.4, 3.5) do not give expected information.The ultimate goal in the issue of determining the deviation from unitarity would be a methodto compute it just from the initial matrix without necessity of its Hermitian product VV † , whichis closely related to the problem of the entrywise characterization of contractions. This togetherwith other issues brought up by us should result in a deeper understanding of the structure of thephysical region Ω . 5 atrix norms, neutrino mixings Wojciech Flieger
Acknowledgements
We would like to thank Janusz Gluza and Marek Gluza for useful remarks and careful readingof the manuscript. Work is supported in part by the COST Action CA16201 and the Polish NationalScience Centre (NCN) under the Grant Agreement 2017/25/B/ST2/01987.
F.P. is supported by theUniversity of Silesia program "Uniwersytet dla Najlepszych" 2018/2019.
References [1] M. Sorel, J. M. Conrad, M. H. Shaevitz, Combined analysis of short-baseline neutrino experiments inthe ( + ) and ( + ) sterile neutrino oscillation hypotheses, Phys. Rev. D 70 (2004) 073004. doi:10.1103/PhysRevD.70.073004 .URL https://link.aps.org/doi/10.1103/PhysRevD.70.073004 [2] G. Karagiorgi, Z. Djurcic, J. M. Conrad, M. H. Shaevitz, M. Sorel, Viability of ∆ m ∼ sterileneutrino mixing models in light of MiniBooNE electron neutrino and antineutrino data from theBooster and NuMI beamlines, Phys. Rev. D 80 (2009) 073001. doi:10.1103/PhysRevD.80.073001 .URL https://link.aps.org/doi/10.1103/PhysRevD.80.073001 [3] J. Kopp, M. Maltoni, T. Schwetz, Are There Sterile Neutrinos at the eV Scale?, Phys. Rev. Lett. 107(2011) 091801. doi:10.1103/PhysRevLett.107.091801 .URL https://link.aps.org/doi/10.1103/PhysRevLett.107.091801 [4] K. N. Abazajian, et al., Light Sterile Neutrinos: A White Paper. arXiv:1204.5379 .[5] C. O. Dib, C. S. Kim, S. Tapia Araya, Search for light sterile neutrinos from W ± decays at the LHC. arXiv:1903.04905 .[6] S. Gariazzo, C. Giunti, M. Laveder, Y. F. Li, Updated global 3+1 analysis of short-baseline neutrinooscillations, Journal of High Energy Physics 2017 (6) (2017) 135. doi:10.1007/JHEP06(2017)135 .URL https://doi.org/10.1007/JHEP06(2017)135 [7] R. Gandhi, B. Kayser, M. Masud, S. Prakash, The impact of sterile neutrinos on CP measurements atlong baselines, Journal of High Energy Physics 2015 (11) (2015) 39. doi:10.1007/JHEP11(2015)039 .URL https://doi.org/10.1007/JHEP11(2015)039 [8] J. Gluza, T. Jeli´nski, Heavy neutrinos and the pp → ll j j CMS data, Phys. Lett. B748 (2015) 125–131. arXiv:1504.05568 , doi:10.1016/j.physletb.2015.06.077 .[9] J. Gluza, T. Jelinski, R. Szafron, Lepton number violation and "Diracness" of massive neutrinoscomposed of Majorana states, Phys. Rev. D93 (11) (2016) 113017. arXiv:1604.01388 , doi:10.1103/PhysRevD.93.113017 .[10] T. Golling, et al., Physics at a 100 TeV pp collider: beyond the Standard Model phenomena, CERNYellow Report (3) (2017) 441–634. arXiv:1606.00947 , doi:10.23731/CYRM-2017-003.441 .[11] S. Dube, D. Gadkari, A. M. Thalapillil, Lepton-Jets and Low-Mass Sterile Neutrinos at HadronColliders, Phys. Rev. D96 (5) (2017) 055031. arXiv:1707.00008 , doi:10.1103/PhysRevD.96.055031 . atrix norms, neutrino mixings Wojciech Flieger[12] A. Abada, et al., Future Circular Collider: Vol. 1 Physics opportunities http://inspirehep.net/record/1713706/files/CERN-ACC-2018-0056.pdf .[13] S. Antusch, E. Cazzato, O. Fischer, A. Hammad, K. Wang, Lepton Flavor Violating Dilepton DijetSignatures from Sterile Neutrinos at Proton Colliders, JHEP 10 (2018) 067. arXiv:1805.11400 , doi:10.1007/JHEP10(2018)067 .[14] T. A. Mueller, et al., Improved Predictions of Reactor Antineutrino Spectra, Phys. Rev. C83 (2011)054615. arXiv:1101.2663 , doi:10.1103/PhysRevC.83.054615 .[15] G. Mention, et al., Reactor antineutrino anomaly, Phys. Rev. D 83 (2011) 073006. doi:10.1103/PhysRevD.83.073006 .URL https://link.aps.org/doi/10.1103/PhysRevD.83.073006 [16] A. P. Serebrov, et al., The first observation of effect of oscillation in Neutrino-4 experiment on searchfor sterile neutrino. arXiv:1809.10561 , doi:10.1134/S0021364019040040 .[17] J. N. Abdurashitov, et al., Measurement of the response of a Ga solar neutrino experiment to neutrinosfrom a Ar source, Phys. Rev. C73 (2006) 045805. arXiv:nucl-ex/0512041 , doi:10.1103/PhysRevC.73.045805 .[18] C. Giunti, M. Laveder, Statistical Significance of the Gallium Anomaly, Phys. Rev. C83 (2011)065504. arXiv:1006.3244 , doi:10.1103/PhysRevC.83.065504 .[19] C. Athanassopoulos, et al., Evidence for ν µ → ν e oscillations from the LSND experiment at LAMPF,Phys. Rev. Lett. 77 (1996) 3082–3085. arXiv:nucl-ex/9605003 , doi:10.1103/PhysRevLett.77.3082 .[20] A. Bolshakova, et al., Revisiting the ’LSND anomaly’ II: critique of the data analysis, Phys. Rev. D85(2012) 092009. arXiv:1112.0907 , doi:10.1103/PhysRevD.85.092009 .[21] M. Maltoni, T. Schwetz, M. A. Tortola, J. W. F. Valle, Ruling out four neutrino oscillationinterpretations of the LSND anomaly?, Nucl. Phys. B643 (2002) 321–338. arXiv:hep-ph/0207157 , doi:10.1016/S0550-3213(02)00747-2 .[22] A. A. Aguilar-Arevalo, et al., Significant Excess of Electronlike Events in the MiniBooNEShort-Baseline Neutrino Experiment, Phys. Rev. Lett. 121 (22) (2018) 221801. arXiv:1805.12028 , doi:10.1103/PhysRevLett.121.221801 .[23] G. Karagiorgi, M. H. Shaevitz, J. M. Conrad, Confronting the Short-Baseline Oscillation Anomalieswith a Single Sterile Neutrino and Non-Standard Matter Effects. arXiv:1202.1024 .[24] D. Döring, H. Päs, P. Sicking, T. J. Weiler, Sterile Neutrinos with Altered Dispersion Relations as anExplanation for the MiniBooNE, LSND, Gallium and Reactor Anomalies. arXiv:1808.07460 .[25] J. Asaadi, E. Church, R. Guenette, B. J. P. Jones, A. M. Szelc, New light higgs boson andshort-baseline neutrino anomalies, Phys. Rev. D 97 (2018) 075021. doi:10.1103/PhysRevD.97.075021 .URL https://link.aps.org/doi/10.1103/PhysRevD.97.075021 [26] V. Alan Kostelecký, M. Mewes, Lorentz and CPT violation in neutrinos, Phys. Rev. D 69 (2004)016005. doi:10.1103/PhysRevD.69.016005 .URL https://link.aps.org/doi/10.1103/PhysRevD.69.016005 [27] J. S. Díaz, V. A. Kostelecký, Lorentz- and CPT -violating models for neutrino oscillations, Phys. Rev.D 85 (2012) 016013. doi:10.1103/PhysRevD.85.016013 .URL https://link.aps.org/doi/10.1103/PhysRevD.85.016013 atrix norms, neutrino mixings Wojciech Flieger[28] E. Bertuzzo, S. Jana, P. A. N. Machado, R. Zukanovich Funchal, Dark Neutrino Portal to ExplainMiniBooNE excess, Phys. Rev. Lett. 121 (24) (2018) 241801. arXiv:1807.09877 , doi:10.1103/PhysRevLett.121.241801 .[29] P. Ballett, S. Pascoli, M. Ross-Lonergan, U(1)’ mediated decays of heavy sterile neutrinos inMiniBooNE. arXiv:1808.02915 .[30] K. Bielas, W. Flieger, J. Gluza, M. Gluza, Neutrino mixing, interval matrices and singular values,Phys. Rev. D98 (5) (2018) 053001. arXiv:1708.09196 , doi:10.1103/PhysRevD.98.053001 .[31] K. Bielas, W. Flieger, Dilations and Light–Heavy Neutrino Mixings, Acta Phys. Polon. B48 (2017)2213. doi:10.5506/APhysPolB.48.2213 .[32] E. Fernandez-Martinez, M. B. Gavela, J. Lopez-Pavon, O. Yasuda, CP-violation from non-unitaryleptonic mixing, Phys. Lett. B649 (2007) 427–435. arXiv:hep-ph/0703098 , doi:10.1016/j.physletb.2007.03.069 .[33] S. Antusch, C. Biggio, E. Fernandez-Martinez, M. B. Gavela, J. Lopez-Pavon, Unitarity of theLeptonic Mixing Matrix, JHEP 10 (2006) 084. arXiv:hep-ph/0607020 , doi:10.1088/1126-6708/2006/10/084 .[34] Z.-z. Xing, Correlation between the Charged Current Interactions of Light and Heavy MajoranaNeutrinos, Phys. Lett. B660 (2008) 515–521. arXiv:0709.2220 , doi:10.1016/j.physletb.2008.01.038 .[35] Z.-z. Xing, A full parametrization of the 6 × arXiv:1110.0083 , doi:10.1103/PhysRevD.85.013008 .[36] Z. Maki, M. Nakagawa, S. Sakata, Remarks on the unified model of elementary particles, Prog.Theor. Phys. 28 (1962) 870–880. doi:10.1143/PTP.28.870 .[37] M. Kobayashi, T. Maskawa, CP Violation in the Renormalizable Theory of Weak Interaction, Prog.Theor. Phys. 49 (1973) 652–657. doi:10.1143/PTP.49.652 .[38] S. M. Bilenky, S. T. Petcov, Massive Neutrinos and Neutrino Oscillations, Rev. Mod. Phys. 59 (1987)671, [Erratum: Rev. Mod. Phys.60, 575(1988)]. doi:10.1103/RevModPhys.59.671 .[39] D. Steinberg, Computation of Matrix Norms with Applications to Robust Optimization, M.Sc. thesis,Technion, Haifa, Israel (2005).URL [40] F. Nie, H. Huang, X. Cai, C. H. Ding, Efficient and Robust Feature Selection via Joint ℓ https://hal.archives-ouvertes.fr/hal-00975276 [42] V. H. Aguiar, R. Serrano, Slutsky matrix norms: The size, classification, and comparative statics ofbounded rationality, Journal of Economic Theory 172 (2017) 163 – 201. doi:https://doi.org/10.1016/j.jet.2017.08.007 . atrix norms, neutrino mixings Wojciech Flieger[43] R. Sanyal, F. Sottile, B. Sturmfels, Orbitopes, Mathematika 57. doi:10.1112/S002557931100132X .[44] J. Saunderson, P. A. Parrilo, A. S. Willsky, Semidefinite descriptions of the convex hull of rotationmatrices, SIAM Journal on Optimization 25. doi:10.1137/14096339X ..